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<table width="100%" summary="page for HalleyLifeTable"><tr><td>HalleyLifeTable</td><td style="text-align: right;">R Documentation</td></tr></table>

<h2>
Halley's Life Table
</h2>

<h3>Description</h3>

<p>In 1693 the famous English astronomer Edmond Halley studied
the birth and death records of the city of Breslau, which
had been transmitted to the Royal Society by Caspar Neumann.
He produced a life table showing the number of people
surviving to any age from a cohort born the same year. He
also used his table to compute the price of life annuities.
</p>


<h3>Usage</h3>

<pre>data("HalleyLifeTable")</pre>


<h3>Format</h3>

<p>A data frame with 84 observations on the following 4 variables.
</p>

<dl>
<dt><code>age</code></dt><dd><p>a numeric vector</p>
</dd>
<dt><code>deaths</code></dt><dd><p>number of deaths, <i>D_k</i>, among people of age k, a numeric vector</p>
</dd>
<dt><code>number</code></dt><dd><p>size of the population, <i>P_k</i> surviving until this age, a numeric vector</p>
</dd>
<dt><code>ratio</code></dt><dd><p>the ratio <i>P_{k+1}/P_k</i>, the conditional probability
of surviving until age k + 1 given that one had already reached age k, a numeric vector</p>
</dd>
</dl>



<h3>Details</h3>

<p>Halley's table contained only <code>age</code> and <code>number</code>.
For people aged over 84 years, Halley just noted that their total number was 107.
This value is not included in the data set.
</p>
<p>The data from Breslau had a mean of 1,238 births per year: this is the value that
Halley took for the size, <i>P_0</i> of the population cohort at age 0. 
From the data, he could compute the annual mean
<i>D_k</i> of the number of deaths among people aged <i>k</i> for all <i>k &gt;= 0</i>.
From this, he calculated the number <i>P_{k+1}</i> surviving one more year,
</p>
<p style="text-align: center;"><i>P_{k+1} = P_k - D_k</i></p>

<p>This method had the great advantage of not requiring a general census but only knowledge of the
number of births and deaths and of the age at which people died during a few years.
</p>


<h3>Source</h3>

<p>N. Bacaer (2011), &quot;Halley's life table (1693)&quot;, Ch 2, pp 5-10.
In <em>A Short History of Mathematical Population Dynamics</em>,
Springer-Verlag London,
DOI 10.1007/978-0-85729-115-8_2.  Data taken from Table 1.
</p>


<h3>References</h3>

<p>Halley, E. (1693). An estimate of the degrees of the mortality of mankind, drawn from
curious tables of the births and funerals at the city of Breslaw; with an attempt
to ascertain the price of annuities upon lives. 
<em>Philosophical Transactions of the Royal Society, London</em>, 17,
596-610.
</p>
<p>The text of Halley's paper was found at
<a href="http://www.pierre-marteau.com/editions/1693-mortality.html">http://www.pierre-marteau.com/editions/1693-mortality.html</a>
</p>


<h3>See Also</h3>

<p><code>Arbuthnot</code>
</p>


<h3>Examples</h3>

<pre>
data(HalleyLifeTable)
# what was the estimated population of Breslau?
sum(HalleyLifeTable$number)

# plot survival vs. age
plot(number ~ age, data=HalleyLifeTable, type="h", ylab="Number surviving")

# population pyramid is transpose of this
plot(age ~ number, data=HalleyLifeTable, type="l", xlab="Number surviving")
with(HalleyLifeTable, segments(0, age, number, age, lwd=2))

# conditional probability of survival, one more year
plot(ratio ~ age, data=HalleyLifeTable, ylab="Probability survive one more year")


</pre>


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